Union of Sets

The union of two sets $A$ and $B$ is the set of all elements that are in $A$, in $B$, or in both.

• Notation: $A\cup B$
• Definition (Set-Builder Notation):

$$A\cup B=\{x\mid x\in A\mathrm{or}x\in B\}$$

• Example:
  • $A=\{1,2,3\},B=\{3,4,5\}$
  • $A\cup B=\{1,2,3,4,5\}$

Properties:

• Commutative: $A\cup B=B\cup A$
• Associative: $(A\cup B)\cup C=A\cup (B\cup C)$
• Idempotent: $A\cup A=A$
• Identity: $A\cup \varnothing =A$